Confusion with the direction of E-fields near conductors

[rudransh verma]

In this page you can see it’s written: E must be perpendicular to the surface. If it were not then the charges would move due to a component along the surface.
I am assuming the field is generated due to the charges on the surface.
I have a doubt. Is the author saying if it were not perpendicular then some charges would leave the surface.
Because maybe the field is not actually perpendicular. That’s why the surface charge density is not same everywhere. Some of the field vectors are not perpendicular and due to that some charges shift from their place and we get uneven charge density.

I don’t understand one more line below: we assume that the cap area A is small enough so that the field Magnitude E is constant…
So that means Es magnitude is changing over the entire surface because of the varying charge density. Some places there is more charge so E produced is more and vice versa. The area A we are considering has the same charge density everywhere in it and so is E in that area.

 

[ergospherical]

The tangential component of the electric field is preserved at an interface between two media, so if the electric field inside the conductor is $0$ then the tangential component of the electric field outside the conductor must be...

 

[rudransh verma]

an interface between two media

What does that mean? I didn’t catch you. Also Please complete your sentence.

 

[Ibix]

What does that mean?

An interface between two media is the surface where two different materials meet.

Please complete your sentence.

The idea is that you learn how to do that. If the electric field is zero inside a conductor and the tangential component doesn't change at the interface, what is the tangential component immediately outside the conductor?

 

[rudransh verma]

An interface between two media is the surface where two different materials meet

Two media means one is air and other is the conductor itself. And interface is the surface of the conductor.

 

[rudransh verma]

The tangential component of the electric field is preserved at an interface between two media, so if the electric field inside the conductor is 0 then the tangential component of the electric field outside the conductor must be...

@Ibix is he saying that there is a tangential component of external field just outside the surface of conductor if it’s not perpendicular.

 

[Ibix]

@Ibix is he saying that there is a tangential component of external field

No, I'm asking you to deduce the value of the tangential component from what I (and @ergospherical) have said.

 

[rudransh verma]

No, I'm asking you to deduce the value of the tangential component from what I (and @ergospherical) have said.

I guess Esintheta

 

[ergospherical]

cmon put some effort into it lad

 

[rudransh verma]

cmon put some effort into it lad

All I know is the perpendicular component would be Ecos and other would be Esin if E is at an angle with area vector.

 

[DaveE]

If you have a tangential e-field component at the surface of a conductor, that will exert a force on the electrons at the surface that will make them move. They will continue to move until their contribution to the total e-field cancels the original applied field. Like a ball rolling into a valley, it will find an equilibrium (ignoring strange cases, like huge fields...). That equilibrium has no force on the electrons, which means no net tangential e-field component. One of the key points here is the assumption that a conductor has an essentially unlimited supply of conduction band electrons that can be easily moved around. Then they will move to a place of rest where they have eliminated the tangential e-field gradients that would otherwise exert a force on them. This doesn't work for the perpendicular component because the electrons aren't free to move beyond the interface (into or out of the conductor).

 

[Ibix]

I guess Esintheta

If the field inside the conductor is zero, what is the tangential field just inside the conductor?

If the tangential field is continuous across the interface, given your answer to the first question, what is the tangential field at the interface?

 

[rudransh verma]

If the field inside the conductor is zero, what is the tangential field just inside the conductor?

Zero ! So?

 

[rudransh verma]

They will continue to move until their contribution to the total e-field cancels the original applied field.

I don’t get that line. There is no applied field. Just the field of positive charges distributed unevenly on the surface of the conductor.
But I can understand that they will find equilibrium somewhere. I am guessing it will not leave the surface.

 

[Ibix]

Zero ! So?

So the field is perpendicular.

 

[rudransh verma]

So the field is perpendicular.

Are you implying that the tangential field just inside is also a tangential field just outside. And since the just inside tangential field is zero so is just outside tangential field.

 

[DaveE]

I don’t get that line. There is no applied field. Just the field of positive charges distributed unevenly on the surface of the conductor.
But I can understand that they will find equilibrium somewhere. I am guessing it will not leave the surface.

I guess I was presupposing the problem of what happens when an externally generated e-field is applied to a conductor. The field at the surface is a combination of the field created by external charges plus the field created by the internal charges.

The case of no applied e-field is the trivial case, then the electrons will move in the conductor to create an e-field with zero gradient inside. What happens at the surface is that any excess charge (like if you had more electrons than protons) will accumulate at the surface evenly distributed, since they repel each other and try to move as far away as possible. But there will still be no tangential field by the same reasoning, if there was, the charges would move along the surface to compensate.

 

[rudransh verma]

The tangential component of the electric field is preserved at an interface between two media

That means same tangential field will have an effect just as much outside just as inside.

 

[ergospherical]

Yeah, well the idea is that if you take a tiny rectangular curve $\mathcal{C}$ straddling the interface then $\displaystyle{\oint}_{\mathcal{C}} \mathbf{E} \cdot d\mathbf{r} = \int_{\mathcal{S}} \nabla \times \mathbf{E} \cdot d\mathbf{S}$. If $\partial_t \mathbf{B}$ is bounded then as $\mathcal{C}$ shrinks the surface integral goes to zero and therefore $\mathbf{E}_1 \cdot \delta \mathbf{l} - \mathbf{E}_2 \cdot \delta \mathbf{l} = \Delta \mathbf{E} \cdot \delta \mathbf{l} = 0$ (where $\delta \mathbf{l}$ is one of the long edges of the rectangle parallel to the interface). Since the orientation of $\delta \mathbf{l}$ in the plane is arbitrary, $\Delta \mathbf{E}$ has no component parallel to the interface.

 

[hutchphd]

What happens at the surface is that any excess charge (like if you had more electrons than protons) will accumulate at the surface evenly distributed, since they repel each other and try to move as far away as possible.

This assumes that the surface has a uniform curvature. As soon as there is a hill or valley the surface charge can be wildly uneven because the direction of tangent changes

 

[rudransh verma]

Yeah, well the idea is that if you take a tiny rectangular curve $\mathcal{C}$ straddling the interface then $\displaystyle{\oint}_{\mathcal{C}} \mathbf{E} \cdot d\mathbf{r} = \int_{\mathcal{S}} \nabla \times \mathbf{E} \cdot d\mathbf{S}$. If $\partial_t \mathbf{B}$ is bounded then as $\mathcal{C}$ shrinks the surface integral goes to zero and therefore $\mathbf{E}_1 \cdot \delta \mathbf{l} - \mathbf{E}_2 \cdot \delta \mathbf{l} = \Delta \mathbf{E} \cdot \delta \mathbf{l} = 0$ (where $\delta \mathbf{l}$ is one of the long edges of the rectangle parallel to the interface). Since the orientation of $\delta \mathbf{l}$ is arbitrary, $\Delta \mathbf{E}$ has no component parallel to the interface.

So what if there is a non perpendicular field due to charges at the surface. That would mean there is a tangential field just inside the conductor. But we know field is zero there. So the field is perpendicular at the surface. Also if it were not true some charges would move around the surface.

 

[Ibix]

That would mean there is a tangential field just inside the conductor.

Then there would be a force on the charges and they would move until there was no force.

 

[rudransh verma]

Then there would be a force on the charges and they would move until there was no force.

And it is possible that they would never stop and continuously move along the surface.

 

[DaveE]

And it is possible that they would never stop and continuously move along the surface.

No, for static fields, there will be an equilibrium state. For externally generated time varying fields, then yes, they will (may?) never stop.

This site has some good illustrations. https://courses.lumenlearning.com/p...rs-and-electric-fields-in-static-equilibrium/

 

[rudransh verma]

No, for static fields, there will be an equilibrium state. For externally generated time varying fields, then yes, they will (may?) never stop.

This site has some good illustrations. https://courses.lumenlearning.com/p...rs-and-electric-fields-in-static-equilibrium/

But the uneven distribution of charge will change and will attain a new uneven distribution. Right?

 

[DaveE]

But the uneven distribution of charge will change and will attain a new uneven distribution. Right?

Yes, but it will be stable. They will stop moving.

 

[rudransh verma]

Yes, but it will be stable. They will stop moving.

Is this field always perpendicular to the surface? Because if it were not then there would be a tangential field just inside. And we know field inside is zero always.

 

[Ibix]

And it is possible that they would never stop and continuously move along the surface.

No, because they will move to reduce their energy and there is eventually some minimum point of that. Or you can observe that electrons will move away from regions of too-negative charge towards regions of too-positive charge, reducing the negativity of the region they leave and the positivity of the region they approach.

The exception, as @DaveE says, is time-varying electric fields where the electrons have to move to keep the field zero - that's how an antenna works.

 

[DaveE]

Is this field always perpendicular to the surface? Because if it were not then there would be a tangential field just inside. And we know field inside is zero always.

Yes. That's what nearly every reply has said.

A tangential field would cause charges to move along the surface, which they are free to do.

The perpendicular e-field at the surface will try to make charges move towards or from the surface. But those charges aren't free to exit the conductor, so they accumulate at the surface. This will cause the internal e-field to be zero, but the external perpendicular field can't be canceled out since the charges can't move across the boundary.

 

[rudransh verma]

@DaveE To summarise I will say there is a isolated conductor which has zero E inside and outside. Now it’s charged. For a brief moment there will be a net internal electric field due to all these charges but that net field will soon disappear because these charges will soon redistribute itself in such a way that the net field on each charge due to all charges is zero. Field at every point become zero. Now the charges are at the surface making perpendicular field to the surface. Why? Its true that the outside tangential field will have an effect inside tangentially. But that doesn’t mean outside field is perpendicular. The field can well be at an angle and there would be an internal tangential field. Why are we saying because the internal field is zero! Of course It’s will be zero . All the charges are now on surface. Real reason is that the charges don’t move on the surface. That’s why the field lines are perpendicular to the surface.
I guess it was experimentally seen that charges don’t move on the surface. They become static. That is why we say fields are perpendicular.

@ergospherical You say inside field is zero. Tangential field both inside and thus outside is zero. Field is so perpendicular. But isn’t it possible that there is no field inside but the field due to surface charge is not perpendicular and have a tangential component along the surface.